To determine the 99% confidence interval for the population proportion of mice who can run a maze in 30 seconds or less, let's go through the detailed steps:
1. Identify the sample size and sample proportion:
- Sample size ([tex]\( n \)[/tex]) = 50 mice
- Sample proportion ([tex]\( \hat{p} \)[/tex]) = 38% (which needs to be converted to decimal for calculations, so [tex]\( \hat{p} = 0.38 \)[/tex])
2. Determine the confidence level:
- Confidence level = 99%, which indicates a high level of certainty that the true population proportion lies within the calculated interval.
3. Calculate the standard error of the sample proportion:
- The formula for the standard error of a proportion is:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
Plugging in the values, we get:
[tex]\[
SE = \sqrt{\frac{0.38 \cdot (1 - 0.38)}{50}} \approx 0.06864400920692205
\][/tex]
4. Find the z-score corresponding to the 99% confidence level:
- The z-score for a 99% confidence level (which corresponds to 0.5% in each tail of the normal distribution) is approximately 2.5758.
5. Calculate the margin of error:
- The margin of error (ME) is calculated by multiplying the z-score by the standard error:
[tex]\[
ME = 2.5758 \cdot 0.06864400920692205 \approx 0.17681525042827034
\][/tex]
6. Determine the confidence interval:
- The confidence interval is found by adding and subtracting the margin of error from the sample proportion:
[tex]\[
\text{Lower bound} = \hat{p} - ME = 0.38 - 0.17681525042827034 \approx 0.20318474957172967
\][/tex]
[tex]\[
\text{Upper bound} = \hat{p} + ME = 0.38 + 0.17681525042827034 \approx 0.5568152504282704
\][/tex]
7. Convert the bounds back to percentages:
- The lower bound in percentage is approximately:
[tex]\[
0.20318474957172967 \cdot 100 \approx 20.318474957172967 \%
\][/tex]
- The upper bound in percentage is approximately:
[tex]\[
0.5568152504282704 \cdot 100 \approx 55.68152504282704 \%
\][/tex]
Therefore, the 99% confidence interval for the population proportion of mice who can run the maze in 30 seconds or less is approximately 20.3% to 55.7%. This means we can be 99% confident that the true population proportion lies within this interval.
So, the correct option from the given choices is:
[tex]\[ \boxed{20.3 \% < p < 55.7 \%} \][/tex]
